Theory and Applications of Primal Weak Structures

This paper presents the notion of primal weak structures as a generalized mathematical framework obtained by relaxing the axioms of classical topological spaces. We formally define primal weak structures and provide a detailed investigation of their fundamental properties. Particular attention is given to the operator cw⋄, whose essential characteristics and related properties are analyzed to obtain a comprehensive characterization of primal weak structures. Furthermore, we introduce new constructions denoted by σ(w,P), π(w,P), β(w,P), and α(w,P), and demonstrate that they generate generalized primal topological spaces. These results establish a unifying connection between primal weak structures and existing generalized topological frameworks. In addition, several separation axioms are proposed to distinguish between different classes of primal weak structure spaces. Overall, primal weak structures constitute a flexible and robust class of mathematical models with strong connections to classical topology and significant potential for future applications. The operators and constructions developed in this work provide a solid foundation for further research in this area.